1.1 Systems of Linear Equations

A linear equation in the variables $x_1,…,x_n$ is an equation that can be written in the form

A linear system is consistent if it has either one or infinitely many solutions. Otherwise, it is inconsistent if it has no solution.

Example: Find values of h such that the matrix is the augmented matrix of a consistent linear system.

A linear system is consistent if it has one or infinitely many solutions. Mathematically, we find a matrix to be inconsistent if it has a row where all the entries are zeros except for the b-entry. We see if this is the case by row-reducing the matrix.

We find that this matrix can’t possibly be inconsistent.

1.2 Row Reduction and Echelon Forms

Echelon Form (REF)

• Leading entries in descending order

Reduced Echelon Form (RREF)

• All leading entries are $1$

• Each leading $1$ is the only nonzero entry in its column

Theorem 1: Uniqueness of the Reduced Echelon Form

Each matrix is row equivalent to one and only one reduced echelon (RREF) matrix

Solutions of Linear Systems

Once a system is in RREF, you can write a general solution form as a linear combination of the free variables.

Example: Find the general solution for $\left[ \begin{matrix} 1 & -2 & 0 \\ 0 & 0 & 1 \\ \end{matrix} \left| \, \begin{matrix} -4 \\ -7 \\ \end{matrix} \right. \right]$

We find that $x_2$ is the free variable, thus we write the general solution to the system in terms of the free variable.

Questions

Find the general solution for $\left[ \begin{matrix} 1 & -2 & -1 \\ 3 & -6 & -2 \\ \end{matrix} \left| \, \begin{matrix} 3 \\ 2 \\ \end{matrix} \right. \right]$

First, we row reduce to RREF.

Then write as a general solution, which I’ve already done in the example above.

1.3 Vector Equations

Span

The set of all linear combinations of $v_1,\dots, v_p$

Questions

Determine if $b$ is a linear combination of $a_1$, $a_2$, $a_3$.

This is equivalent to asking if $b$ is in $Span(a_1,a_2,a_3)$. Just solve for a solution $x$ for $Ax=b$ where the columns of $A$ are the vectors $a_i$.

1.4 The Matrix Equation $Ax=b$

The equation $Ax=b$ has a solution if and only if $b$ is a linear combination of the columns of $A$.

Questions

For a matrix $B \in R^{4\times4}$, do the columns of $B$ span $R^4$? Does the equation $Bx=y$ have a solution for each $y$ in $R^4$?

This is equivalent to asking if the columns of $B$ are linearly independent.

1.5 Solution Sets of Linear Systems

The homogeneous equation $Ax=0$ has a nontrivial solution if and only if the equation has at least one free variable (linearly dependent).

Theorem 6: Solutions of Nonhomogeneous Systems

The solution for $Ax=b$ is equivalent to the solution $Ax=0$ transposed by some vector $p$.

Questions

Write the solution set of the following homogeneous system in parametric vector form.

Row-reduce, write the general form solution in parametric vector form:

Describe the solutions of $Ax=0$ in parametric vector form for $A= \begin{bmatrix}1 & -2 & -9 & 5\\ 0 & 1 & 2 & -6\end{bmatrix}$

Row reduce into RREF:

General solution:

Find the parametric equation of the line $a= \begin{bmatrix}3 \\ -4\end{bmatrix}$ parallel to $b= \begin{bmatrix}-7 \\ 8\end{bmatrix}$

The line parallel to $b$ is given by $b$ itself, to make sure this line is through the point $a$, you shift it by $a$

Find a parametric equation of the line $M$ through $p$ and $q$

The line that goes runs parallel to $p$ and $q$ is given by $p-q$. To make sure it actually goes through $p$ and $q$, we just add either vector to it

1.7 Linear Independence

An indexed set of vectors $\{v_1,\dots,v_p\} \in R^n$ is linearly independent if the vector equation

has only the trivial solution $x = \vec 0$.

A set of two vectors $\{v_1, v_2\}$ is linearly dependent if one of the vectors is a multiple of the other.

Theorem 7: Characterization of Linearly Dependent Sets

An indexed set $S=\{v_1,\dots,v_p\} \in R^n$ is linearly dependent iff at least one of the vectors in $S$ is a linear combination of the others.

Theorem 8

If a set $S=\{v_1,\dots,v_p\} \in R^n$ contains more vectors than there are entries in each vector ($p> n$), then the set is linearly dependent. Matrices that are horizontally rectangular have linearly dependent column vectors.

Theorem 9

If a set $S=\{v_1,\dots,v_p\} \in R^n$ contains the zero vector, then the set is linearly dependent because the equation $x_1v_1 + x_2v_2 + \dots + x_pv_p = \vec 0$ has a non-trivial solution.

Questions

True or False: The columns of $A\in\R^{4\times5}$ are linearly dependent.

True by Theorem 8.

True or False: If $x$ and $y$ are linearly independent, and $\{x,y,z\}$ is linearly dependent, then $z$ is in $Span\{x,y\}$

True. There exists some combination $c_1x+c_2y=c_3z$

5, 9, 11, 17, 25, 27.

1.8 Introduction to Linear Transformations

A transformation (or function or mapping) $T$ from $R^n$ to $R^m$ is a rule that assigns to each vector $x$ in $R^n$ a vector $T(x)$ in $R^m$.

• $R^n$ is the domain of $T$

• $R^m$ is the codomain of $T$

• $colA$ is the range of $T$

Matrix Transformations

Example: Let $A \in R^{3\times2}$, $u\in R^2$, $b ,c \in R^3$.

• Find $T(u)$, the image of $u$ under $T$.

• Find an $x\in R^2$ whose image under $T$ is $b$

  Just solve $Ax=b$

• Is there more than one $x$ whose image under $T$ is $b$?

  This is equivalent to asking if $Ax=b$ has infinitely many solutions. Row-reduce $A$ and see if it has a free variable.

• Is $c$ in the range of $T$?

  Check if $Ax=c$ has a solution.

Linear Transformations

Given that a linear transformation is defined by a matrix transformation $x \mapsto Ax$, it has the following properties:

Additionally:

Onto

One-to-One

Questions

Let $A\in R^{4\times 3}$, $b\in R^4$, $T(x) = Ax$. Find a vector $x$ whose image under $T$ is $b$.

Just solve $Ax=b$

Describe geometrically what $T$ does to each vector $x$ in $R^2$. $T(x) = \begin{bmatrix}0 & 1 \\ 1 & 0 \end{bmatrix}\begin{bmatrix}x_1\\x_2 \end{bmatrix}$.

It flips the $x$ and $y$ values, resulting in a reflection along $\begin{bmatrix}1 \\ 1\end{bmatrix}$

True or False: A linear transformation preserves the operations of vector addition and scalar multiplication

True by definition.

True or False: Every matrix transformation is a linear transformation

True. Ax is closed under vector addition and scalar multiplication.

1.9 The Matrix of a Linear Transformation

Theorem 10

A linear transformation $T(X)$ is defined by a unique matrix $A$ such that

Onto

If each $b \in \R^m$ is the image of at least one $x \in \R^n$, the mapping is onto. $Ax=b$ has at least one solution for each $b$ (spans the codomain) if and only if $\text{Rank}A = m$

One to One

If each $b \in \R^m$ is the image of at most one $x \in \R^n$, the mapping is one to one. $Ax=b$ has exactly one solution for each $b$.

Theorem 11

$T$ is one to one if and only if the $T(x)=0$ has only the trivial solution.

Theorem 12

• $T$ is onto iff the columns of $A$ span $\R^m$

• $T$ is one-to-one iff the columns of $A$ are linearly independent

Example: Is the linear transformation defined by A onto or one-to-one?

The matrix is full rank, thus it spans the codomain, so we know that it is onto.

By Theorem 11 we know that $T$ is only one-to-one if the columns of $A$ are linearly independent. By Theorem 8, we know that when there are more columns than rows, the columns are automatically linearly dependent, so it’s not one-to-one.